Let's break down the given expression step by step:

Expression:

$\cos\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}(\sqrt{3})\right)$

Step 1: Solve $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$

We need to find the angle $\theta_1$ such that: $\sin(\theta_1) = \frac{\sqrt{3}}{2}$ From basic trigonometry, we know that: $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ So, $\theta_1 = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

Step 2: Solve $\cot^{-1}(\sqrt{3})$

We need to find the angle $\theta_2$ such that: $\cot(\theta_2) = \sqrt{3}$ Since $\cot(\theta) = \frac{1}{\tan(\theta)}$, we can use the fact that: $\cot\left(\frac{\pi}{6}\right) = \sqrt{3}$ So, $\theta_2 = \cot^{-1}(\sqrt{3}) = \frac{\pi}{6}$

Step 3: Combine the angles

Now we can add the two angles together: $\theta_1 + \theta_2 = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}$

Step 4: Find the cosine

Now, we can compute the cosine of $\frac{\pi}{2}$: $\cos\left(\frac{\pi}{2}\right) = 0$

Final Answer:

$\cos\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}(\sqrt{3})\right) = 0$

Let me know if you would like more details or have any other questions!

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Here are 5 related questions to explore further:

1. What is the relationship between the sine and cotangent functions?
2. How does the range of inverse trigonometric functions affect the angle values?
3. What are the basic identities for the inverse trigonometric functions?
4. Can you solve trigonometric equations involving multiple inverse functions?
5. How do you convert between radians and degrees in trigonometry?

Tip: Always make sure to understand the fundamental identities of trigonometric functions, especially for the inverse ones, as they help simplify complex expressions.